Question

级数证明求解2

见图片

请写出详细的推理过程，谢谢

Answer 1

(2n+3)/n(n+1)=a/n+b/(n+1)=(an+a+bn)/n(n+1)
a+b=2,a=3,b=-1
(2n+3)/n(n+1)=a/n+b/(n+1)=3/n-1/(n+1)
(1->N)∑ (2n+3)/n(n+1) *(1/3)^(n+1)=
(1->N)∑(1/3)^(n)/n-(1->N)∑1/(n+1)*(1/3)^(n+1)=
(1->N)∑(1/3)^(n)/n-(2->N+1)∑(1/3)^(n)/n=
=1/3-(1/3)^(N+1)
(1->∞)∑(2n+3)/n(n+1) *(1/3)^(n+1)=lim (N->∞) 1/3-(1/3)^(N+1)=1/3